K. Benmeriem, C. Bouzar
The aim of this paper is to prove that the well known non solvable Mizohata type partial differential equations have Colombeau generalized solutions which are distributions if and only if they are solv- able in the space of Schwartz distributions. Therefore the Colombeau generalized solvability includes both a new solution concept and new mathematical objects as solutions.
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We present a differential algebra of generalized functions over a field of generalized scalars by means of several axioms in terms of general algebra and topology. Our differential algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions, and the set of regular distributions with $\mathcal C^\infty$-kernels forms a differential subalgebra. We discuss the uniqueness of the field of scalars as well as the consistency and independence of our axioms. This article is written mostly to satisfy the interest of mathematicians and scientists who do not ne…
In distribution theory the pullback of a general distribution by a $C^{\infty}$-function is well-defined whenever the normal bundle of the $C^{\infty}$-function does not intersect the wavefront set of the distribution. However, the Colombeau theory of generalized functions allows for a pullback by an arbitrary c-bounded generalized function. It has been shown in previous work that in the case of multiplication of Colombeau functions (which is a special case of a $C^{\infty}$ pullback), the generalized wave front set of the product satisfies the same inclusion relation as in the distributional …
In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau' theory of new generalized functions and its applications. The main purpose of our non-standard approach to Colombeau' theory is the improvement of the properties of the scalars of the varieties of spaces of generalized functions: in our non-standard approach the sets of scalars of the functional spaces always form algebraically closed non-archimedean Cantor complete fields. In contrast, the scalars of the…
We study a model elliptic pseudo-differential equation and simplest boundary value problems for a half-space and a special cone in Sobolev--Slobodetskii spaces which have different smoothness with respect to separate variables. Sufficient conditions for a unique solvability for such boundary value problems are described.
In this article we investigate the solvability of infinite-dimensional differential algebraic equations. Such equations often arise as partial differential-algebraic equations (PDAEs). A decomposition of the state-space that leads to an extension of the Hille-Yosida Theorem on Hilbert spaces for these equations is described. For dissipative partial differential equations the famous Lumer-Phillips generation theorem characterizes solvability and also boundedness of the associated semigroup. An extension of the Lumer-Phillips generation theorem to dissipative differential-algebraic equations is …
We set-up and solve the Cauchy problem for Schrödinger-type differential operators with generalized functions as coefficients, in particular, allowing for distributional coefficients in the principal part. Equations involving such kind of operators appeared in models of deep earth seismology. We prove existence and uniqueness of Colombeau generalized solutions and analyze the relations with classical and distributional solutions. Furthermore, we provide a construction of generalized initial values that may serve as square roots of arbitrary probability measures.
We consider the Cauchy problem for a hyperbolic pseudodifferential operator whose symbol is generalized, resembling a representative of a Colombeau generalized function. Such equations arise, for example, after a reduction-decoupling of second-order model systems of differential equations in seismology. We prove existence of a unique generalized solution under log-type growth conditions on the symbol, thereby extending known results for the case of differential operators with generalized functions as coefficients.
We present an extension of the methods of classical Lie group analysis of differential equations to equations involving generalized functions (in particular: distributions). A suitable framework for such a generalization is provided by Colombeau's theory of algebras of generalized functions. We show that under some mild conditions on the differential equations, symmetries of classical solutions remain symmetries for generalized solutions. Moreover, we introduce a generalization of the infinitesimal methods of group analysis that allows to compute symmetries of linear and nonlinear differential…
We tackle the problem of finding a suitable categorical framework for generalized functions used in mathematical physics for linear and non-linear PDEs. We are looking for a Cartesian closed category which contains both Schwartz distributions and Colombeau generalized functions as natural objects. We study Frölicher spaces, diffeological spaces and functionally generated spaces as frameworks for generalized functions. The latter are similar to Frölicher spaces, but starting from locally defined functionals. Functionally generated spaces strictly lie between Frölicher spaces and diffeological s…
From zero solutions with partially bounded supports to solvability and Runge approximation for partial differential equations
研究齐次偏微分方程具规定支撑性质非平凡解的存在性，为常系数偏微分算子建立有部分有界支撑的光滑零解存在性。通过多种方法获偏微分方程可解性与逼近相关结果，包括\(P\)-凸性特征、龙格逼近等，还开发龙格逼近框架并应用于相关系统。
Solvability of invariant systems of differential equations on $\mathbb{H}^2$ and beyond
本文利用非紧型对称空间上向量丛分布截面的 Fourier 变换，类比 Hörmander 的证明方法，研究不变微分方程系统的可解性问题，并证明了双曲平面的完全可解性及其他双曲空间的部分可解性结果。
On the solvability of boundary value problems for linear differential-algebraic equations with constant coefficients
本文通过参数化方法和 Weierstrass 标准型，给出了常系数线性微分代数方程两点边值问题唯一可解性的判据。
Using the existence of infinite numbers $k$ in the non-Archimedean ring of Robinson-Colombeau, we define the hyperfinite Fourier transform (HFT) by considering integration extended to $[-k,k]^{n}$ instead of $(-\infty,\infty)^{n}$. In order to realize this idea, the space of generalized functions we consider is that of generalized smooth functions (GSF), an extension of classical distribution theory sharing many nonlinear properties with ordinary smooth functions, like the closure with respect to composition, a good integration theory, and several classical theorems of calculus. Even if the fi…
We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of {\em Colombeau type} in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. The article is aimed at mathematicians and physicists who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theor…
Nearness and solvability of non-invariant equations on stratified groups
本文利用 Campanato 算子邻近性理论，证明分层群上含非不变、非线性二阶算子的方程 $Au=f$ 的适定性，并推广至任意维 Heisenberg 群及 Euclid 无界区域。
Solvability of Infinite Systems of Nonlinear Caputo Fractional Differential equations in generalized Hahn Sequence Space
本文在广义Hahn序列空间中，利用非紧性Hausdorff测度和Banach不动点定理，研究非线性Caputo分数阶微分方程无限系统的解的存在性、唯一性及Hyers-Ulam稳定性，并通过实例验证结果。
We construct a diffeomorphism invariant (Colombeau-type) differential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing differential calculus in infinite dimensional (convenient) vector spaces, previous attempts in this direction are unified and completed. Several classification results are achieved and applications to nonlinear differential equations involving singularities are given.
We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers w…
The boundary-value problem on semi-axis for one class operator-differential equations of the fourth order, the main part of which has the multiple characteristic is investigated in this paper in Sobolev type weighted space. Correctness and unique solvability of the boundary-value problem is proved, and the solvability conditions are expressed in terms of the operator coefficients of the equation. Estimations of the norms of the operators of intermediate derivatives, closely connected with the solvability conditions, have been carried out. The connection between the exponent of the weight and t…
A fairly general continuation theorem of Leray-Schauder type for the class of so-called admissible multimaps is set forth. This result is then used to establish a universal rule for solving operator inclusions of Hammerstein type in Lebesgue-Bochner spaces. Examples illustrating the legitimacy of this approach include the initial value problem for perturbation of $m$-accretive mutivalued differential equations, the anti-periodic problem for semilinear differential inclusions, abstract integral inclusions of Fredholm and Volterra type and the two-point boundary value problem for nonlinear evolu…
An Embedding of Schwartz Distributions in the Algebra of Asymptotic Functions
本文通过将分布空间嵌入到一种名为“渐近函数”的微分代数中，解决了 Schwartz 分布的乘法问题，该代数与 Colombeau 代数相似但不同。
We analyze solvability of a special form of distributed order fractional differential equations within the space of tempered distributions supported by the positive half-line.
This paper lays the foundations for a nonlinear theory of differential geometry that is developed in a subsequent paper which is based on Colombeau algebras of tensor distributions on manifolds. We adopt a new approach and construct a global theory of algebras of generalised functions on manifolds based on the concept of smoothing operators. This produces a generalisation of previous theories in a form which is suitable for applications to differential geometry. The generalised Lie derivative is introduced and shown to commute with the embedding of distributions. It is also shown that the cova…
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